We highlight this potential by means of an illustrative and practically relevant example: predicting expectation values of Pauli operators in an unknown \(n\) -qubit quantum state \(\rho \) .
This is a central task for many quantum computing applications [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
To formulate this problem in our framework, suppose the \(2n\) -bit input \(x\)  specifies one of the \(4^n\)  \(n\) -qubit Pauli operators \(P_x \in \lbrace I, X, Y, Z\rbrace ^{\otimes n}\) , and suppose that \(\operatorname{{\mathcal {E}}}_{\rho }(|x\rangle \!\langle x|)\)  prepares the unknown state \(\rho \)  and maps \(P_x\)  to the fixed observable \(O\) , which is then measured; hence
\(f(x) = \operatorname{tr}(O \operatorname{{\mathcal {E}}}_\rho (\vert x \rangle \! \langle x \vert )) =\operatorname{tr}(P_x \rho ).\) 
